That could open the door for Chelsea to make a £25million bid for Torino defender Nikola Maksimovic, who is also wanted by Napoli and last week went on strike to try to force through a transfer, but AC Milan have warned the Blues off a move for Alessio Romagnoli.

Koulibaly emerged as Chelsea head coach Antonio Conte’s first-choice defensive target over the past month, but the Blues have failed with a series of bids for the 25-year-old as Napoli have so far held firm.

Torino defender Nikola Maksimovic (R) is another Chelsea target

The last offer is understood to have been around £40m and Chelsea have informed Napoli that they are unwilling to significantly increase that bid ahead of the closure of the transfer window next Wednesday.

With time running out and Conte determined to sign a new defender, Chelsea are now working through their list of alternatives while they wait to find out whether or not Napoli will relent.

Napoli want Koulibaly to sign a new contract with the club, but it is understood the player would prefer to move to Stamford Bridge this month.

It remains to be seen whether or not Chelsea’s interest in Maksimovic changes Napoli’s thinking. Keeping an unhappy player with the threat of missing out on their first-choice replacement could leave the Italian club in a difficult position in the long run.

Leonardo Bonucci will stay at Juventus

Conte has been keen to recruit a new defender from Serie A, where he knows the market best from his time as Juventus manager and as head coach of the Italian national team, but Chelsea have found the market difficult.

Having quickly realised that Leonardo Bonucci could not be prised away from Juve, Conte targeted Koulibaly only to find that Napoli want huge money for the Senegal international.

Italian sources claim an enquiry for 21-year-old Romagnoli has been rejected by AC Milan, which leaves Chelsea short on options in Italy aside from Maksimovic and Lazio’s Stefan de Vrij.

On the future of Maksimovic, who has been compared to his fellow Serb Nemanja Vidic and Juventus star Leonardo Bonucci, Torino director of sport Gianluca Petrachi said: “We’ve been talking to Napoli about Maksimovic since January, so there’s been too much talk and not enough action.

“Now he is wanted by a big foreign club too, but at the same time we won’t rule out him staying too. It’s not just Napoli who want Maksimovic, but also a big club from abroad.”

Asked if the big club from abroad is Chelsea, Petrachi added: “It is an English club and the Premier League does have greater financial resources than other leagues.”

Having already signed striker Michy Batshuayi and midfielder N’Golo Kante, Conte had hoped to secure another four players by the closure of the August transfer window.

Portugal goalkeeper Eduardo is expected to sign from Dinamo Zagreb, but, while the club remain active in the market, Chelsea are finding it tough to fulfil all of Conte’s requests this month.

Other than spending the next nine days trying to boost Conte’s squad, Chelsea face a busy period of offloading a host of players either permanently or on loan.

Juventus want to take Juan Cuadrado back to Italy on loan, but Chelsea would ideally prefer a permanent deal, while Kenedy, who has been offered to Watford, Matt Miazga, Patrick Bamford, Lucas Piazon and Christian Atsu are among the players who will be loaned out.

Marko Marin, who joined Chelsea four years ago but started only two Premier League games for the club, has joined Olympiakos on a permanent deal.

Having failed to qualify for Europe, Chelsea face Bristol Rovers in the second round of the League Cup on Tuesday night at Stamford Bridge with Batshuayi, Ruben Loftus-Cheek and Ola Aina in line to make their first starts of the season.